Wave filter



R. L DxETzoLD WAVE FILTER Fild June 9, 1954 Fla/- ATTORNEY Sept. 22,1936.

Patented Sept. 22, 1936 WAVE FILTER Robert L. Dietzold, New York, N. Y.,assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y.,a corporation of New York Application June 9, 1934,` Serial No. 729,7334 claims. (ci. ris-,44)

This, vinvention relates to frequency selective networks and moreparticularly to the control of the phase characteristics of broad bandselective systems.

`5 It has for its principal object the provision of a linear phasecharacteristic not only in the transmission band of a band selectivesystem, but also .through the band limits and into the attenuationranges as far as maybe desired.

` characteristic which is made linear in this way is transfer constantof the lter network but alsov the `wave-reflection effects at theterminals andthe overall characteristic of the filter network incombination with its terminal impedances. which in practice willgenerally be fixed resistances. 3 This overall characteristic involvesnot only the the'desi'red linearity is obtained as the'result of theproper coordination of the sum total of these 1 reilection'effectswiththe transfer characteristic 'u of the lter network.

Y The nature ofthe invention will be more fully understood fromthefollowing detailed description and from the accompanying drawing ofwhich: Y

Fig. 1 shows schematically a general type of network of the invention; c

Fig. 2 is a reactance characteristic used in the explanation of theinvention; u

Figs. 3 and 4 illustrate vtlfiecharacter of the impedances in aparticular embodiment of the network of Fig. l; and

Fig. 5 illustrates certain characteristics of the networks ofthe'inventio'n.- i

` Referringftoj Fig'. 1, the network illustrated comprises a symmetricallattice having series and diagonal impedances Za and Zb respectively,connected between equal terminal resistances, R in series Vwith' one ofwhich-is a Wave source E. The branch impedances may be 4of any degree ofcomplexity;A but should be' substantially free `from dissipation.

The properties of the symmetrical lattice are described at length inUnited States Patent t 1,828,445,4,-issuedpctober,20, 1931130 H. IW.Bode This object is attained vby the use in the.

wherein it is shown that by particular allocation of the resonance andanti-resonance frequencies of the branch impedances, hereinafterdesignated critical frequencies, certain advantageous frequencycharacteristics of the image impedance and the transfer constant may beprovided. The characteristics discussed in the Bode patentV are those ofthe lattice per se, namely, its image impedance and transfer constant,asdistinguished from the overall properties of the lattice plus theimpedances between which it is connected. In thelatter case thetransmission characteristic Vof the system is not represented by thetransfer constant alone but by this factor together with modifyingfactors representing the reection effects at the junctions of thelattice and the eX- ternal impedances. The'total effect, which is ameasure of the ratio of the currents at the re` ceiving end of thesystem before and after the' insertion of the lattice is termed theinsertion transfer factor.

The present invention is concerned with the phase component of theinsertion transfer factor, that is, with the sum of al1 the phase shiftsin the system including those produced by reflection effects. The termsinsertion phase shift and insertion phase characteristic are used todesignate the phase component of the insertion transfer factor. Inaccordance with the invention this phase shift is made to have a linearvariation' with frequency not only within the transmission band but alsothrough the attenuating ranges by Va particular allocation of thecritical frequencies of the branch impedances. This allocation is suchthat, except at each side of the cut-off frequencies, the criticalfrequencies are separated by a uniform interval both in the transmissionband and in the attenuating ranges, the separation at each side of thecut-olf frequencies being reduced to three quarters of the intervalelse- Where.

The analysis which follows is directed to the demonstration of thelinearity of the phase characteristic obtained by the simple frequencyarrangement of the invention and tothe determination of the latticebranch impedancesk so that the network will exhibit this linearitycouple with band selective properties.

The determination is further subject tothe-f condition that theimpedance arms be physically-V s PATENT OFFICE realizable. Thiscondition imposes a certain functional form for the dependence upon the,fr quency, which may be quickly ascertained. The insertion transferfactor for the network is conveniently examined in terms of the imagetransfer constant, 0, and the image impedance, Z1, Which are related tothe lat-tice impedances, Za and Zt, say by the equations Z tan h 2- Zb(l) ZI=J (2) Equation (l) shows that for free transmission, or for 0 apure imaginary, Za/Zb must be negative. This result is achieved over anarbitrary frequency interval if Za and Zb are reactances unlike in sign,that is, reactances of which the alternating resonances andanti-resonances correspond, a resonance in Za to an anti-resonance in Zband so son. Also,by Equation (l) ,the network attenuates in a frequencyinterval in which Za/Zb is positive, for then 0 is real. This ensues ifZa and Zh are alike in sign, or if resonances in Za correspond toresonances in Zb, and so for antiresonances. Since a condition for thephysical realizability of a reactance is that its resonances andanti-resonances alternate, between an interval of transmission and aninterval of suppression there must occur a critical frequency, thecut-off, in one impedance arm only.

The impedances Za and Zt by their frequency variations and magnitudescompletely determine the transmission properties of the network and forthat reason may be termed characterizing impedances.

Therefore, filter properties: are obtainable from a physicallyrealizable lattice network if only the arms are reactances having theappropriate type of correspondence between their respective naturalfrequencies. This is illustrated in the case of the low-pass` lter, inwhich the branch impedances Za and Zb are of the types shown in Figs. 3and 4 respectively, by the reactance ex- 1Where Ka and Kb are constantsand where f1 and f2 are critical frequencies in the transmitting band,fc a cut-01T intermediate between f2 and f3, and f4 and f5 criticalfrequencies in the attenuating band. For convenience the criticalfrequencies representing resonances are termed zeros and thoserepresenting anti-resonances are termed poles. Within the transmissionband of a filter the zeros and poles are inversely coincident, that is,the zeros of the one impedance are coincident with the poles of theother impedances, while in the attenuation ranges the zeros and thepoles of the two impedances are directly coincident, zeros with zeros Y'and poles With poles. Plots of the impedances, showing the manner ofcoincidence of the resonant frequencies, are given by Fig. 2 in whichfull line curve l0 represents the frequency variation of the reactanceof Za and dotted line curve Il represents the variation of Z. With thesevalues for Za and Zh, the Equations (1) and (2) become It is seen that 0is imaginary and Z1 real for f f, whereas for f f, 0 is real and Z1imaginary, corresponding tothe case of the low-pass filter. It willexpedite the discussion to confine the attention to this case,subsequently extending the results tol high-pass and band-pass filters.

The relations (la) and (2a) then indicate the form which the dependenceof the image parameters upon the frequency must take in orderv that thenetwork may be a physically realizable low-pass filter. Evidently norestriction is placed upon the number of transfer-constant controllingfrequencies (fi and f2 in the example) Vnor upon the number of impedancecontrolling frequencies (f3 and f4 in the example). The cut-off factor,

f2 f2 Y (Jn-E2) n (lfaZn-l) f2 wel) el "fr (1 3) f2 Y I fbx,

wherein K1 and K2 are constant real quantities. The solution of (1b) and(2b) always yields physically realizable expressions for Za and Zb ifthe Ks are positive and The adjustment of the lattice elements accord-The insertion constant of the network is de fined by e"'Y=I-r, Where Irand Ir are the received currents before and after theinsertion of thenetwork.v When expressed in terms of the image parameters and theterminating resistance, R, fy is found to be a sum of the transferconstant and the reflection and interaction constants. These latter aredeiinedrespectively by eh: *T2 and en: 1 Te (4) (1+ -I c Y R It may benoted in passing that the interaction constant defined by. Equation 4represents the repeated reflection of the initially reflected part ofthe current or wave as it passes back and forth betweenY the terminal.impedances and infinite number of times. The convenience of this form ofexpression becomes manifest when one eX- amines the variation in thephase shift separately in the three intervals, the transmitting band,the attenuating band, and the transition band. In so doing isestablished the distribution of the critical frequencies j.,i and f icorresponding to linear phase shift.

Transmitting band-From Equation (2b) it is seen that tends to 1 as ftends toward zero, if'K2 be taken equal to R.. Furthermore, the form ofthe function is such that Z1 differs but little from R in this interval,the immediate vicinity of the cutoff having been set aside for thetransition inter-l val. On this account in the pass band thecontributions of the reflection'and interaction factors to the phaseshift are negligible and the transfer constant represents substantiallythe total insertion loss. This is readily seen from Equations (3) and(4), the right-hand sides of which converge to the value unity as Z1approaches the value R, corresponding to negligibly small values of thereflection and interaction constants r and Y 0i. Since this conditionholds throughout the interval in question, thel phase characteristicthere is determined substantially wholly by the transfer constant alone.If 6=a|7`, where i is the imaginary unit, then by (1b),

when A increases by vrias f varies from one critical frequency to thenext. In order thatthe slope be constant throughout the interval, it istherefore necessary that the Vcritical frequencies fm be uniformlyspaceds If this spacing is Af, then the phase shift undulates about thechord" Lr i x 2 2 Af having its ideal value at least'at each criticalfrequency. Y Y n VAttenuatimy band- In this interval,v 'Ithe imaginarypart of the transfer constant is either Zero or 1?; whileinteraction'effects are negligible on account ofthe factor e26 in (4)',With 0 real. The part of the phase-shift dependent-upon-the frequency istherefore the imaginary part of the reflectionconstant, 0r. Since Z1 yisreactive in this range, the phase of the denominator of (3) isA whilethat of the numerator is a 2 arctan 1. R

Thus

6,: :Fi-+2 arctan 1R (6) The significance of the constant term willappear presently. With the help of (2b), the second term is seen to be,in the attenuating band, a function of the same type as the transferconstant in the transmitting band. Hence, for the phase slope to beconstant in the attenuating range, the impedance controlling factorsalso must constitute a chain of uniformly spaced resonances andanti-resonances. Since r increases by 1r between successive criticalfrequencies,l the slope will be equal to the slope in the pass band ifthe uniform spacing is the same constant Af in both ranges.

Transition band-It remains to determine the frequency spacings adjoiningthe cut-off so that the phase curves in the transmitting'and.attenuating bands are joined through the transition band by a chord ofthe same slope. In this interval, which we suppose to be bounded by theVlast uniformly spacedY critical frequencies in the transfer constant andimpedance controlling chains and to contain only the cut-olf frequency,neither the reflection nor interaction effects are negligible. In fact,for this method of decomposing the total insertion loss, thesecomponents become op-positely infinite at the cutfoff. kHowever, theinteraction factor introduces no net change of phase over the interval,since it vanishes at one edge in virtue of Z1 equal to R very nearly andat the other in virtue of e2e being very small. It may therefore beignored in evaluating the total change in phase through this interval.

In the space adjoining the cut-off on the transmitting side, the phaseof. the transfer constant increases by 1f. Inthe space adjoining thecutoff on the attenuating side, the phase `of the reflection factorincreases by 1r. At the cut-off, however, where the image impedancechanges from real to imaginary, the reflection factor in-Y troduces 'anabrupt change in phase of jto represented by the first term of (6).Therefore the net change in the transition interval is radians, and theinterval must contain 3/2 uniform spaces if the average slope is to becorrect. Considerations of symmetry require that the cutoif be thecenter of the interval, which thus comprises two three-quarter spaces.

These observations establish necessary conditions upon the frequencypattern corresponding to the requirement of linear phase shift in bothtransmitting and attenuating bands. The sunlciency of these conditions,when appropriate values have been assigned to K1 and K2 in Equations(lb) and (2b), may be verified by direct computation. For this purposethe formulae for the reflection and interaction factors are not usefulbecause of the indeterminacy at the cut-off. This difficulty is avoidedby expressing Z1 and 0 in terms of the lattice impedances, in whichevent If y'Xa and jYb be written for Za and Zh, the insertion loss andphase shift, Afy and B'y, are given n n R R eAT= and

tan B'y which is increased by 1r radians in the threequarter interval onthe attenuation side of the cut-off. The contribution of the interactionconstant must, of course, remove this phase discontinuity at thecut-off. In the pass band, its imaginary part is where Z1 is real. Thelimiting value of this angle as the cut-off is approached throughfrequencies in the pass band can be determined by' expressing it interms of the lattice impedances. At the cut-off, either Xa or Xb iseither zero or infinite. Suppose that Xa is zero. Then this limit isarctan On the attenuation side of the cut-off, the imaginary part of theinteraction constant is a i arc [Pe 2 lttee 1R] where Z1 is imaginary.The limiting value of this expression, as the cut-off is approachedthrough frequencies in the attenuation band, is found by the same methodto be Xb arctan )16: fc.

Thus the discontinuity in l at the cut-off is asV required to remove thediscontinuity at this point introduced by the reflection effect.

The variation with frequency ofthe several phase shift components isillustrated by the curves of Fig. 5 for the case of the low-pass filterhaving impedances Za and Zh in accordance with Figs. 3 and 4respectively, and having the critical frequencies spaced in the mannerdescribed. CurveV I2 represents the transfer phase shift that is, thephase component yof the transfer constant of the lattice per se, in thetransmission range from zero frequency to the cut-o. This componentincreases by 1r in each of the intervals between the criticalfrequencies, including fc, and undulates about the straight line I5departing therefrom by 1r/4 at the cut-olf. In the figure, theundulations of this curve, as well as those of the other curves aresomewhat exaggerated in order that their character may be exhibited.

Curve I3 represents the reflection phase shift r in the attenuatingrange, this component being zero in the transmission band. By Virtue ofthe critical frequency spacing the general slope of this curve is thatof the line I5 but it is characterized first by a departure of 1r/4 atthe cutoff and a sudden change of 1r at the critical frequency f3.amounts to a reversal of phase its effect in general is not material.

Curve I4 represents the interaction phase shift. This curve ischaracterized by undulations of half the period of those of the othercurves and by a sudden change of 1r/2 at the cut-off.

The total phase shift in the system is obtained by adding the threecurves togetherin which case it will be noted that the discontinuity ofcurve I4 at the cut-off just neutralizes that at th-e junction of curvesI2 and 32. The resultant phase shift will therefore show a smoothvariation which is very close to linear through the whole range fromzero to ,f3 and which continues at the same slope, subject to reversalsat the critical frequencies, in the higher range.

The pattern for the transfer constant and impedance controllingfrequencies which has been found is suflicient to insure only that thephase shift has its linear value at each critical frequency, or that theaverage slope in each space be the same. In order that the slope mayclosely approximate to the average at every intermediate point, it isfurther necessary to `determine the multipliers K1 and K2 of thetransfer constant and image impedance expressions. We have already seenthat K2 should be taken equal tothe terminating impedance, R, so as toobtain impedance match and vanishing interaction effects in the passband. K1 may be evaluated from Since this latter change simply Equationof which the -principal part in the limit of smallf is i gf-af Y Y n Thechord with which the phase characteristic should coincide is 5 1 11 2- 2Af whenceV 7l' Kl-f f2, must be uniformly spaced, falling at Af and'2M.A The cut-off, fc, is separated from f2 by three-quarters a uniforminterval, and from the first cf the uniformly spaced impedancecontrolling frequencies by a like interval. Thus, Equations (1a) and(2a) Vbecome (Peitz) in which Af may be selected toY bring the cut-offto any desired point on the frequency scale. The solutions of theserelations for the lattice impedances are then The element values for theimpedances are readily found by expanding these expressions in partialfractions, after the manner described by R. M. Foster, A reactancetheorem, Bell System Technical Journal, v. 3, No. 2, April, 1924.

With this choice of, parameters the greatestV deviation of the phaseslope from the average is found by` computation to be of the order of lper cent'. This approximation is satisfactory for mostpracticalpurposes. Since 'all the parameters of the network have been determinedwith an eye to the phase characteristic, this is accompanied by a uniqueloss characteristic. The loss characteristic is marked by refiectionpeaks at each impedance controlling frequency, where the latticeimpedances are zero or infinite together. At these frequencies the imageimpedance changes sign, and therefore also the constant term of Equation(6). Thus, although the phase slope is uniform throughout theattenuating range, the phase characteristic itself has discontinuitiesof 1r radians at each impedance controlling frequency. 'Ihis is Ytheinterpretation of the constant term of Equation (6). Whether this is anincrease or a decrease of 1r radians is not distinguishable foranon-dissipative network. When parasitic `dissipation of energyin'the'network elements is taken into account, the reiiection peaks ofloss have finite maxima and the phase in the neighborhood increases ordecreases by 1r according as the lineor cross-arm of the lattice has thesmaller resistance component at the peak frequency. The infinite peak atthis frequency, and the associated abrupt change in phase, can evidentlybe restored by adding a lumped resistance to the smaller impedance so asto bring the arms into balance. This observation is of importance inconsidering the effect of dissipation on the phase shift.

When the network is constructed of physical elements its performancecharacteristics will be somewhat changed from those computed upon theassumption of pure reactance lattice arms. However, the relationssubsisting'` between the real and imaginary parts of any analyticfunction such as the insertion constant enable these changes to be,readily computed so long as the dissipation can be regarded asuniformly distributed among the elements. In fact, if d is the averageratio of resistance to reactance in the elements, and AA and AB are thevariations in the insertion loss and phase shift due to the introductionof dissipation, we have approximately DB Armada (9) and oA wd-; (10)where the derivatives are computed for the network of pure reactancesThe frequency variable w is 27d. Now the dissipation is ordinarilyconcentrated chiefly in the coils, so thatk wd is constant if the coilresistances are constant. Then the effect of dissipation upon the losscharacteristic in a linear phase shift network is simply the addition ofa uniform loss. Y

Moreover, throughout the transmitting band,

in which there is by Equation (1'0)V no first order change in the phasecharacteristic. But in the transition interval, when the loss isincreasing, the phase curve is displaced through dissipation from theideal straight line. This effect may be compensated in two ways. Itdepends upon the dissipation being uniformly distributed among theresonant combinations of which the network,

is composed, and is modified if that distribution is modified. Inparticular if lumped resistance be added to the meshes resonating at thefirst impedance controlling frequency in such a way as to balance thelattice at this frequency, the phase curve will be restored tolinearity, as predicted above.

The effect of dissipation on the phase characteristic in the transitioninterval Amay be otherwise corrected for by small variations in theide-al frequency pattern. By diminishing slightly the two three-quarterspace intervals in the transition band, the non-dissipative phase slopemay be caused progressively to increase through the band so that changedue to parasitic dissipation displaces the characteristic toward, ratherthan away from, the ideal straight line. Since to shorten the cut-offspacing increases the selectivity of the network, the lattenuationcharacteristic is improved by increase of dissipation in the impedancestogether with compensating modification of the frequency spacing in thisway. The appropriate variations of vthe critical frequencies from theirtheoretical locations are best determined by trial.

It is possible in other ways to obtain a measure of control over theloss characteristic by means of slight variations in the ideal values ofthe parameters. For eXample, the loss may be increased at the cost ofsome degradation of the phase Vproperty by varying the constants K1 andK2.

Since the spacing of impedance controlling frequencies must be uniformover that portion of the attenuating band in which the phase slope is tobe uniform, the extension of this condition over the infiniteattenuating band of a lowpass filter would result in an infinitenetwork. In practice the phase slope is seldom of interest very far intothe attenuating band, so that the chain of uniformly spaced impedancecontrolling frequencies may be soon terminated. If the phase requirementends at a frequency fd, uniform spacing must be maintained through fd.Then the infinite chain of uniformly spaced critical frequencies greaterthan fd may be replaced by one or more critical frequencies so locatedthat the corresponding factors approximate in the range below fd to thefactors associated with the omitted infinite sequence. The numericaldetermination of the terminating critical frequencies is simple, since aclose approximation is obtained by use of one, or at most two, of themat somewhat extended spacings.

The foregoing discussion has for simplicity been confined to the -caseof the low-pass lter. Similar observations may be made in respect toband-pass and high-pass filters. For the bandpass filter we must have achain of uniformly spaced critical frequencies in the pass band withcut-offs at three-quarter spacing at both edges. Uniform spacing ofimpedance controlling frequencies in both attenuating bands is resumedafter three-quarter intervals beyond the cutoffs. Since the lowercut-off factor replaces the factor,

ofthe transfer constant expression in the frequency range above thelower cut-off, the theoretical constant multiplier is unity. Themultiplier of the image impedance expression is determined to make theimpedance R at the mean of the cut-E frequencies.

The high-pass filter may be regarded as the limiting case of theband-pass filter as the upper cut-off recedes toward infinity. Thepreservation of linear phase shift over this infinite pass band wouldrequire an infinite network on account of the necessity of uniformspacing of transfer constant controlling frequencies, but if there is afrequency, fd, beyond which the phase shift is not of interest, thehigh-pass filter may be realized in a nite'netwcrk by terminating thechain of critical frequencies beyond this point in the manner describedabove for impedance controlling frequencies.

What is claimed is:

1. In a broad band selective system comprising a symmetrical reactancenetwork having multiple resonant characterizing impedances Za. and Zh,and equal resistive terminal impedances connected to the input and theoutput terminals of the network, the method of producing a linear phaseshift throughout the band and beyond the limits thereof which comprisesspacing the critical frequencies of the characterizing impedances atuniform intervals throughout the greater portion of the transmissionband and in a-portion of an attenuation range beyond a band limit andspacing the critical frequencies on each side of said band limit atintervals therefrom substantially equal to three-quarters of the uniforminterval elsewhere.

2. A broad band selective system comprising a symmetrical four-terminalreactance network having characterizing impedances Za and Zb, and equalresistive terminal impedances connected to the input and the outputterminals of said network, said characterizing impedances each having aplurality of critical frequencies which are spaced at uniform intervalsthroughout the greater portion of the transmission band and in anattenuation range beyond a cut-off frequency, and which on each side ofthe cut-off frequency are spaced at intervals substantially equal tothree-quarters of the uniform spacing elsewhere whereby the insertionphase characteristic is linear throughout the band and a portion of theattenuation range.

3. A broad band selective system comprising a symmetrical four-terminalreactance network having characterizing impedances Za and Zb,

and equal resistive terminal impedances connected to the input and theoutput terminals of said network, said characterizing impedances havinga plurality of critical frequencies certain of which lie within thetransmission band and others of which lie outside the band and one ofwhich determines a band limit, said critical frequencies being spaced atuniform intervals throughout the transmission band and in a portion ofthe attenuation range beyond said band limit and having a spacing oneach side of said band limit substantially equal to three-quarters ofthe uniform .spacing elsewhere whereby the insertion phasecharacteristic of the network is a substantially linear Yfunction of thefrequency throughout the band and through the cut-off frequency.

4. A system in accordance with claim 3 in which the poles and zeros ofthe impedance Za are inversely coincident with the poles and zeros ofthe impedance Zb within the band and are directly coincident with thepoles and zeros of impedance Zh outside the band.

ROBERT L. DIETZOLD.

